CN107612553B - Signal index regeneration nuclear sparse sampling method with random pulse positions - Google Patents

Signal index regeneration nuclear sparse sampling method with random pulse positions Download PDF

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CN107612553B
CN107612553B CN201710874582.XA CN201710874582A CN107612553B CN 107612553 B CN107612553 B CN 107612553B CN 201710874582 A CN201710874582 A CN 201710874582A CN 107612553 B CN107612553 B CN 107612553B
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宋寿鹏
申静静
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Jiangsu University
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Abstract

The invention provides a signal index regeneration kernel sparse sampling method with any pulse position. Aiming at the problems that when the existing index regeneration kernel sampling method is used for sparse sampling and reconstruction of signals, the signal reconstruction is inaccurate or even completely incapable due to the fact that the pulse position in the signals appears in a specific interval, the method is suitable for index regeneration kernel sampling of the limited-length signals at any pulse position. The constraint of a reconstruction algorithm on the pulse position of a signal in the existing index regeneration nuclear sampling method is converted into the constraint on a sampling interval, and the sampling of any signal at the pulse position can be realized under the condition of not increasing the number of sampling points by re-determining the sampling interval, so that the problem that the existing index regeneration nuclear sampling method cannot meet the requirement of directly carrying out sparse sampling on an actual signal with random and unadjustable parameters is solved, and a theoretical basis is provided for realizing sparse sampling by hardware.

Description

Signal index regeneration nuclear sparse sampling method with random pulse positions
Technical Field
The invention belongs to the field of signal sampling, and particularly relates to a finite-length signal index regeneration nuclear sparse sampling method with any pulse position.
Background
The Nyquist sampling theorem states that the original signal can be recovered without distortion only if the signal sampling frequency is not less than twice the maximum frequency of the signal. However, with the continuous development of sampling techniques, conventional signal acquisition and storage techniques are inadequate when faced with increasing information requirements and wider signal frequency bands. In order to break through the constraint of bandwidth on the sampling Rate, an fri (complete Rate of innovation) sampling framework is developed, which essentially transforms a specific signal meeting sparsity in an analog domain by using a designed kernel function, so that a small amount of sampled data can also retain useful information enough for reconstructing the signal. The sampling rate in this framework is determined by the degrees of freedom required for signal characterization, which is much lower than conventional Nyquist sampling.
In the FRI sparse sampling framework, the current mainstream sampling kernels include Sinc kernel, gaussian kernel, polynomial regeneration kernel, exponential regeneration kernel, and sos (sum of sincs) kernel. Due to infinite time domain support, the Sinc kernel and the Gaussian kernel have complex and unstable reconstruction algorithm when applied to FRI sampling of finite-length signals; the polynomial regeneration core and the exponential regeneration core have the same framework and belong to regeneration sampling cores, the polynomial regeneration core is a special case of the exponential regeneration core, and therefore the degree of freedom of parameter setting of the exponential regeneration core is higher; the SoS sampling core has better reconstruction robustness, but when the SoS sampling core is used for a finite length signal, the signal or the sampling core needs to be subjected to periodic prolongation, direct hardware realization is difficult, and a multipath accurate delay circuit can complicate a system and increase instability. The index regeneration core related by the invention is not only suitable for finite-length signals, but also has flexible and convenient parameter design of the sampling core, and in addition, the frequency domain expression of the index regeneration core can be converted into a circuit system transfer function, thereby being convenient for hardware realization.
However, for the specific signal, i.e. the signal formed by a series of time-shift weighting of known pulses, when sparse Sampling is performed by using the existing exponential regeneration core Sampling method, it is necessary to ensure that the signal pulses fall in a specific interval, otherwise the reconstruction is inaccurate or even completely fails, so at present, the results of the exponential regeneration core theory and simulation research, such as "Sampling strategies and reconstruction signatures of fine rate of information: Shannon media structure-fix." (needle luminescence drive, Martin vetteri, third blue. IEEE transport. Signal Process.2007,55(5): Trau1. through 1757.) and "FRI Sampling transport array kernel." (Jose Antonio U ü, Thiierray, Pieri write. through 533. through 5323. the IEEE simulation parameters are not directly set in the range of IEEE simulation 5321. through further simulation.
However, in actual FRI signal sampling, the positions of pulses in the signal have randomness, so that it is very urgent to fully ensure that the pulses in the signal appear at any position and can be effectively reconstructed by using sparse sampling data, otherwise, the application of the sampling theory in practice is severely restricted.
Disclosure of Invention
The invention provides an index regeneration nuclear sparse sampling method which can be suitable for any signal at a pulse position, and aims to solve the problems that the pulse position of the existing index regeneration nuclear sampling method is limited and the sparse sampling of the actual signal with random and unadjustable parameters cannot be directly carried out. According to the method, the sampling interval is determined again, the accurate signal reconstruction of the pulse at any position can be effectively guaranteed under the condition that the number of sampling points is not increased, and a theoretical basis is provided for hardware to realize sparse sampling.
For the convenience of describing the present invention, the original signal and the exponential regeneration kernel are briefly introduced:
1) the original signal x (t), i.e. the signal formed by a series of time-shifted weights of known pulses, can be fully characterized by a finite number of information degrees of freedom, which can be expressed as:
Figure BDA0001417834990000021
wherein tau is the signal duration, η (t) is the known pulse, K is the number of pulses, the three are the prior parameters before signal sampling, the amplitude and the time delay corresponding to the K pulses
Figure BDA0001417834990000022
The signal characteristic parameter is sparsely sampled to be measured, and the signal can be uniquely determined.
The existing sampling method has certain limit to the pulse position of the original signal, but the invention is provided for the signal with any pulse position, namely tk∈[0，τ]。
2) Exponential regeneration nucleus
Figure BDA0001417834990000026
I.e. a type of function that can reproduce an index by a shift weighted sum. Has the following characteristics:
Figure BDA0001417834990000023
where M is the order of the exponential regeneration nucleus, αmNuclear parameters were exponentially regenerated.
Figure BDA0001417834990000024
Index of reproducibility
Figure BDA0001417834990000025
cm，kNamely the corresponding index regeneration coefficient.
The technical scheme of the invention is as follows:
an exponential regeneration nuclear sparse sampling method with any pulse position is specifically shown as the attached figure 1, and comprises the following steps:
step 1Determining an exponential regeneration kernel
Figure BDA0001417834990000027
The order M;
step 2, determining sparse sampling parameters including the number N of sampling points and a sampling interval T;
step 3, determining index regeneration core
Figure BDA0001417834990000037
Parameter αm，m＝1，2，…，M；
Step 4, after the original signal x (t) is introduced into a sampling system h (t), outputting y (t) ═ x (t) × h (t);
step 5, sampling y (T) at low speed and equal intervals by time T to obtain sparse sampling value yn，n＝0，1，…，N-1；
Step 6, sparse sampling value ynEstimating parameters of an original signal
Figure BDA0001417834990000038
Finally reconstructing to obtain pulse signals
Figure BDA0001417834990000039
In the above scheme, the index regeneration core is specifically required in the step 1
Figure BDA00014178349900000310
The order M should satisfy M is larger than or equal to 2K, and K is the pulse number of the original signal.
In the foregoing solution, the sparse sampling parameter determining process in step 2 includes:
1) the number N of sampling points satisfies that N is more than or equal to M + 1;
2) the sampling interval T is determined by the number M of sampling cores, the number N of sampling points and the signal tau, and the specific determination method comprises the following steps:
Figure BDA0001417834990000031
the specific sampling process of the existing sampling method is in the signalCollecting N points within a long period of tau, i.e. the sampling interval is based on
Figure BDA0001417834990000032
To be determined. Under the method, the tail part of the signal necessarily has an interval L (tau-MT, tau), and if a pulse occurs in the interval, the reconstruction of the signal is inaccurate and even the signal completely fails, namely t is requiredk∈[0，τ-MT]。
The method provided by the invention is characterized in that the sampling interval is determined, and the accurate reconstruction can be ensured when the position of the signal pulse is not limited under the condition that the number of sampling points is not changed due to different methods for determining the sampling interval.
In the above scheme, the index regeneration core in step 3
Figure BDA00014178349900000311
Parameter α1，α2，…，αMThe requirements are as follows:
1)αmm is 1, 2, …, M has an equi-differential form, i.e. can be expressed as
Figure BDA0001417834990000033
2) To satisfy that the index regeneration kernel is a true kernel, α is requiredmWhere M is 1, 2, … and M is a real number or a conjugate, α0σ + j ω, i.e. requirement
3) Require that
Figure BDA0001417834990000035
For an arbitrary original signal of pulse position, tk∈[0，τ]Then it should satisfy
Figure BDA0001417834990000036
4) In addition, the index regeneration nuclear parameter α is selectedmWhile the above conditions are satisfied, it is still necessary to satisfy:
Figure BDA0001417834990000041
in the above scheme, the sampling system h (t) in step 4 is determined according to the following formula:
compared with the prior sampling method in the form of exponential regeneration kernel inversion, i.e. in a non-causal mannerAs a sampling system, the sampling system determined by the method is a causal system, so that the actual sampling process can be intuitively reflected, and the analysis of actual application is facilitated.
In the above scheme, the specific implementation process of step 5 includes: and (4) collecting the integral points N-M at equal intervals in the signal duration tau at the time interval T for the output signals y (T) in the step (4), and then collecting the M points determined by the sampling core order at equal intervals in a delayed manner.
In the above scheme, the specific implementation process of step 6 includes:
1) firstly, let ynAnd index regeneration coefficient cm，-nLinear combination to obtain:
Figure BDA0001417834990000044
2) coefficient of utilization
Figure BDA0001417834990000045
For parameter smCorrecting to obtain parameter s'm：
s′m＝sm/Δm
3) According to the formula
Figure BDA0001417834990000046
Estimating parameters of original signal by using spectrum estimation method
Figure BDA0001417834990000047
4) By estimating parameters
Figure BDA0001417834990000048
And reconstructing the pulse waveform η (t) to obtain a pulse signal
Figure BDA0001417834990000049
The principle of the invention is as follows:
for the acquired sparse sample values:
Figure BDA0001417834990000051
it is mixed with an index regeneration coefficient cm，-mThe linear combination yields:
Figure BDA0001417834990000052
if smSatisfies the following conditions:
Figure BDA0001417834990000053
then there is
Figure BDA0001417834990000054
In the form of a weighted sum of power series, in which case the parameters
Figure BDA0001417834990000055
The estimation process of (a) can be translated into a general spectrum estimation problem.
Regeneration of nuclei according to index
Figure BDA0001417834990000056
Regeneration index interval R [ (M-N) T, T ] in the case of finite time shift { N | N ═ 0, 1, …, N-1}]I.e. of the formula
Figure BDA0001417834990000057
Want smSatisfy the aboveThe condition, the requirement signal satisfies:
Figure BDA0001417834990000058
the sampling interval determined by the existing sampling method isThen the signal needs to satisfy:
Figure BDA00014178349900000510
it can be seen that, in this method, there must be an interval L ═ (τ -MT, τ) at the tail of the signal, and if a pulse occurs in this interval, the reconstruction process cannot be effectively implemented.
The index regeneration nuclear sparse sampling method with any pulse position provided by the invention determines the sampling interval asAt this time, the signal has no pulse position limited interval, and the pulse can be reconstructed when appearing at any position.
The invention has the beneficial effects that:
aiming at the problem that the pulse position is limited in the existing sampling method, the constraint of the reconstruction algorithm on the pulse position of the signal in the existing sampling method is converted into the constraint on the sampling interval, and the sparse sampling of the signal with the nonadjustable pulse position in the actual situation is realized by re-determining the sampling interval. The method effectively ensures that the pulse can be accurately reconstructed at any position under the condition of not increasing the number of sampling points, and has important significance for effectively applying the index regeneration kernel sampling theory to sparse sampling of actual measurement occasions.
Drawings
FIG. 1 is a flow chart of an exponential regeneration nuclear sparse sampling method with arbitrary pulse positions according to the present invention;
FIG. 2 illustrates an embodiment of a normalized pulse delay parameter of an original signal;
FIG. 3 is a sparse sampling diagram under the prior art exponential regeneration nuclear sampling method according to an embodiment; wherein (a) is a sampling pattern of signal 1; (b) is a sampling graph of signal 2; (c) is a sampling graph of signal 3;
FIG. 4 is a diagram illustrating an effect of estimating reconstruction parameters according to an exemplary conventional exponential regeneration kernel sampling method; wherein (a) is an estimated effect graph of signal 1; (b) is an estimated effect graph of the signal 2; (c) is an estimated effect graph of the signal 3;
FIG. 5 is a sparse sampling diagram under the sampling method of the present invention described in the embodiment; wherein (a) is a sampling pattern of signal 1; (b) is a sampling graph of signal 2; (c) is a sampling graph of signal 3;
FIG. 6 is a diagram illustrating an estimation effect of reconstruction parameters under the sampling method according to the embodiment of the present invention; wherein (a) is an estimated effect graph of signal 1; (b) is an estimated effect graph of the signal 2; (c) is an estimated effect map of signal 3.
Detailed Description
The technical scheme of the invention is further described by combining the drawings and the embodiment as follows:
the exponential regeneration nuclear sparse sampling method with any pulse position provided by the invention is established on the premise that the pulse number K, the pulse shape η (t) and the signal duration tau of an original pulse signal x (t) are prior quantitiesWhere σ represents the width of the gaussian pulse.
In the embodiment, 3 signals with 4 pulse numbers, namely signal 1, signal 2 and signal 3, are selected to respectively carry out signal sparse sampling and reconstruction. For the convenience of analysis, the given time and amplitude parameters respectively perform normalization processing on the actual signal duration and the actual maximum pulse amplitude. Therefore, the signal duration τ is 1, and the relative gaussian pulse width is 0.005. Unknown parameters of 3 signals: normalized amplitude values are all set to
Figure BDA0001417834990000063
Normalized pulse delay parameter
Figure BDA0001417834990000071
As shown in fig. 2.
First, the exponential regeneration kernel sparse sampling and reconstruction process of embodiment signal 1 is described:
step 1: knowing the number K of the signal pulses to be 4, and selecting the sampling core number M to be 8 according to the condition that M is more than or equal to 2K.
Step 2:
1) according to the sampling core order determined in the step 1, N is more than or equal to M +1, and the number of sampling points N is 16 for comparison with the existing sampling method;
2) the number M of sampling core orders, the number N of sampling points and the signal duration tau determined according to the steps are jointly determined, and a formula is determined by the sampling intervalThe corresponding sampling interval T is found to be 0.125.
And step 3: the following conditions are considered comprehensively:
1)αmm is 1, 2, …, M has an equi-differential form, i.e. can be expressed as2) Requirement αmWhere M is 1, 2, … and M is a real number or a conjugate, α0σ + j ω, i.e. requirement
Figure BDA0001417834990000073
3) For arbitrary original signal of pulse position, i.e. tk∈[0，τ]Satisfy the following requirements
Figure BDA0001417834990000074
The sampling kernel parameters are set to λ 0.18j, α 0-0.25-0.81 j.
Due to the fact that for the Gaussian pulse
Figure BDA0001417834990000075
It is always true that the selected sampling kernel parameter λ is 0.18j, α0-0.25-0.81j satisfies the condition.
And 4, step 4: the exponential regeneration kernel takes the most basic E-spline function as an example, then
Figure BDA0001417834990000076
The frequency domain expression is
Figure BDA0001417834990000077
A sampling system according to
Figure BDA0001417834990000078
It is determined that there is tight temporal support [0, MT).
And introducing the original signal into a sampling system to obtain an output signal y (t) ═ x (t) × (t).
And 5: specific sparse sampling process of the output signal y (t) in the step 4: after 8 points are collected at equal intervals within the signal duration tau at the time interval of 0.125, the 8 points determined by the sampling core order are collected at equal intervals in a delayed mode, and 16 points are collected in total.
Step 6: the specific parameter estimation and reconstruction process is as follows, wherein the spectrum estimation algorithm selects a nulling filtering algorithm:
1) firstly, according to the formula
Figure BDA0001417834990000081
Calculating an index regeneration coefficient cm，-nObtaining the sparse sampling value y obtained in the step 5nAnd index regeneration coefficient cm，-nLinear combination to obtain:
Figure BDA0001417834990000082
2) coefficient of utilizationFor parameter smCorrecting to obtain parameter s'm：
s′m＝sm/Δm
3) From s'mM1, 2, …, M constructs the following equation:
Figure BDA0001417834990000084
and solve the unknown coefficient { A }1，A2，…，AK}；
4) To find { A }1，A2，…，AKFilter with coefficientZero point of
Figure BDA0001417834990000086
5) According to the equation
Figure BDA0001417834990000087
Calculating pulse delay parameters
Figure BDA0001417834990000088
6) Will be provided with
Figure BDA0001417834990000089
Substituted type
Figure BDA00014178349900000810
Determining pulse amplitude parameters
Figure BDA00014178349900000811
7) Finally, according to the parameters
Figure BDA00014178349900000812
And the pulse η (t) reconstructs a pulse signal
Figure BDA00014178349900000813
The existing sampling method is different in that: the sampling interval determined in step 2
Figure BDA00014178349900000814
Namely, 16 points are collected at equal intervals in the signal duration tau in the step 5, and the number of the rest index regeneration cores, the parameters and the signal reconstruction process are all consistent. At this time, there is a pulse-limited interval L of (0.5, 1)]If t appearskPulses > 0.5 will result in reconstruction inaccuracies or even complete failure. The sparse sampling process and the reconstruction parameter estimation result are respectively shown in fig. 3 and fig. 4.
The above steps 4, 5, and 6 are repeated for the signal 2 and the signal 3, respectively, and the sparse sampling process and the reconstruction parameter estimation result for obtaining 3 signals are shown in fig. 5 and fig. 6, respectively.
It can be seen that, for the existing sampling method, the estimation result of the signal 1 has a large deviation; only three pulses can be estimated within the duration of signal 2; signal 3, except for the first pulse, all the other estimates are invalid. For the index regeneration nuclear sparse sampling method with any pulse position provided by the invention, the signal can be effectively reconstructed under the condition of not causing extra acquisition data volume by re-determining the sampling interval. The method solves the problem that the existing method is restricted to the pulse position, and has important significance for effectively applying the index regeneration nuclear sampling theory to sparse sampling of actual measurement occasions.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example," or "some examples" or the like mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims (2)

1. An exponential regeneration nuclear sparse sampling method with any pulse position is characterized by comprising the following steps:
step 1, determining index regeneration core
Figure FDA0002287318340000011
The order M;
step 2, determining sparse sampling parameters including the number N of sampling points and a sampling interval T;
step 3, determining index regeneration core
Figure FDA0002287318340000012
Parameter αm，m＝1，2，…，M；
Step 4, after the original signal x (t) is introduced into a sampling system h (t), outputting y (t) ═ x (t) × h (t);
step 5, sampling the signal y (T) at low speed and equal intervals by time T to obtain a sparse sampling value yn，n＝0，1，…，N-1；
Step 6, sparse sampling value ynEstimating parameters of an original signal
Figure FDA0002287318340000013
Finally reconstructing to obtain pulse signals
Figure FDA0002287318340000014
In the step 1, the nucleus is exponentially regenerated
Figure FDA0002287318340000015
The order M is equal to or more than 2K, and K is the number of pulses of the original signal;
the sparse sampling parameter determination process in the step 2 comprises the following steps:
1) the number N of sampling points satisfies that N is more than or equal to M + 1;
2) the sampling interval T is determined by the index regeneration core number M, the number of sampling points N and the signal duration tau, and the specific determination method comprises the following steps:
Figure FDA0002287318340000016
the sampling system h (t) in the step 4 is determined according to the following formula:
Figure FDA0002287318340000017
the sampling method also comprises the steps of converting the constraint of a reconstruction algorithm on the pulse position of the signal in the existing index regeneration core sampling method into the constraint on the sampling interval, and realizing the sampling of any signal at the pulse position by re-determining the sampling interval;
the specific implementation process of the step 5 comprises the following steps: and (4) collecting the integral points N-M of the output signals y (t) in the step (4) at equal intervals in the signal duration tau, and then collecting M points determined by the index regeneration core order at equal intervals in a delayed mode.
2. The method for sparse sampling of exponential regeneration kernels with arbitrary pulse positions according to claim 1, wherein the exponential regeneration kernels in step 3
Figure FDA0002287318340000018
Parameter αmM is 1, 2, …, M is required to satisfy:
1)αmm is 1, 2, …, M has an equi-differential form, i.e. can be expressed as
Figure FDA0002287318340000021
2) To satisfy that the index regeneration kernel is a true kernel, α is requiredmWhere M is 1, 2, …, and M is a real number or a conjugate, when α is given0σ + j ω, i.e. requirement
Figure FDA0002287318340000022
3) Require thatFor an arbitrary original signal of pulse position, tk∈[0，τ]Then it should satisfy
Figure FDA0002287318340000024
4) In addition, the index regeneration nuclear parameter α is selectedmWhile the above conditions are satisfied, it is still necessary to satisfy:
Figure FDA0002287318340000025
wherein
Figure FDA0002287318340000026
For a pulse, σ characterizes the width of the Gaussian pulse;
the specific implementation process of the step 6 comprises the following steps:
1) firstly, sparse sampling value ynAnd index regeneration coefficient cm，-nLinear combination to obtain:
Figure FDA0002287318340000027
2) coefficient of utilization
Figure FDA0002287318340000028
For parameter smCorrecting to obtain parameter s'm：
s′m＝sm/Δm
3) According to the formula:
Figure FDA0002287318340000029
estimating parameters of original signal by using spectrum estimation method
Figure FDA00022873183400000210
4) By estimating parameters
Figure FDA00022873183400000211
And reconstructing the pulse waveform η (t) to obtain a pulse signal
Figure FDA00022873183400000212
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